Figure REF  depicts the spin structures of 40-nm-sized Fe spheres for several defect concentrations \(x_{\mathrm {d}}\)  and at an applied magnetic field of \(\mu _0 H_0 = 0.02\)  T (\(\mathbf {H}_0 \parallel \mathbf {e}_z\) ). While the defect-free case [Fig. REF (a)] exhibits a vortex-type spin configuration (reproducing the results from [1]}), increasing \(x_{\mathrm {d}}\)  results in the progressive disordering (randomization) of the structure [Figs. REF (b) and (c)]. For the here-considered particle size of \(D = 40 \, \mathrm {nm}\) , which is larger than the single-domain size of Fe (\(D_{\mathrm {cr}}^{\mathrm {sd}} \cong 13.6 \, \mathrm {nm}\) ), the vortex structure in Fig. REF (a) is clearly a consequence of the dipolar interaction. Leaving out the dipolar energy in the simulations (i.e., setting \(E_{\mathrm {D}} = 0\) ) results in a quasi single-domain state [Fig. REF (d)]. For \(x_{\mathrm {d}} = 0 \, \%\)  and small applied fields, the vortex structure appears for particle sizes \(D \gtrsim 20 \, \mathrm {nm}\) . Overall, we see that (for \(D = 40 \, \mathrm {nm}\) ) adding defects changes the vortex-type spin structure significantly towards more disordered spin configurations.
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